This number may be reduced drastically by restricting the allowed partitions of \(a_1\) . One simplification is to restrict to flavor symmetries with small numbers of \(U(1)\)  factors—for example, only up to two \(U(1)\)  factors. Another is to restrict to flavor symmetries with only a certain number of simple factors—for example, up to 3 simple factors. Since these two conditions are satisfied by all known rank-two theories [1]}, we find it reasonable to impose them both, though it is certainly possible for there to exist unknown theories violating them. These conditions cut the number of cases down to something of order \(O(10^{12})\) —a large number to be sure, but one which is now accessible with advanced enough computing.
